IPS Chapter 12

© 2010 W.H. Freeman and Company

Inference for On-Way Analysis of Variance

Comparing the Means

One-Way Analysis of Variance

One-way ANOVA Inference for one-way ANOVA

IPS Chapter 12.1

© 2009 W.H. Freeman and Company

Objectives (IPS Chapter 12.1)

Inference for one-way ANOVA

Comparing means

The two-sample t statistic

An overview of ANOVA

The ANOVA model

Testing hypotheses in one-way ANOVA

The F-test

The ANOVA table

The idea of ANOVAReminders: A factor is a variable that can take one of several levels used to

differentiate one group from another.

An experiment has a one-way, or completely randomized, design if several

levels of one factor are being studied and the individuals are randomly

assigned to its levels. (There is only one way to group the data.)

Example: Four levels of nematode quantity in seedling growth experiment.

But two seed species and four levels of nematodes would be a two-way

design.

Analysis of variance (ANOVA) is the technique used to determine

whether more than two population means are equal.

One-way ANOVA is used for completely randomized, one-way designs.

Comparing means

We want to know if the observed differences in sample means are

likely to have occurred by chance just because of random sampling.

This will likely depend on both the difference between the sample

means and how much variability there is within each sample.

Two-sample t statistic

A two sample t-test assuming equal variance and an ANOVA comparing only

two groups will give you the exact same p-value (for a two-sided hypothesis).

H0: =

Ha: ≠

t-test assuming equal variance

t-statistic

H0: =

Ha: ≠

One-way ANOVA

F-statistic

F = t2 and both p-values are the same.

But the t-test is more flexible: You may choose a one-sided alternative instead,

or you may want to run a t-test assuming unequal variance if you are not sure

that your two populations have the same standard deviation .

We first examine the multiple populations or multiple treatments to

test for overall statistical significance as evidence of any difference

among the parameters we want to compare. ANOVA F-test

If that overall test showed statistical significance, then a detailed

follow-up analysis is legitimate.

If we planned our experiment with specific alternative hypotheses in

mind (before gathering the data), we can test them using contrasts.

If we do not have specific alternatives, we can examine all pair-wise

parameter comparisons to define which parameters differ from which,

using multiple comparisons procedures.

An Overview of ANOVA

Do nematodes affect plant growth? A botanist prepares

16 identical planting pots and adds different numbers of

nematodes into the pots. Seedling growth (in mm) is

recorded two weeks later.

Nematodes and plant growth

Nematodes0 10.8 9.1 13.5 9.2 10.65

1,000 11.1 11.1 8.2 11.3 10.435,000 5.4 4.6 7.4 5 5.6

10,000 5.8 5.3 3.2 7.5 5.45

Seedling growth

overall mean 8.03

x i

Hypotheses: All i are the same (H0)

versus not All i are the same (Ha)

The ANOVA model

Random sampling always produces chance variations. Any “factor

effect” would thus show up in our data as the factor-driven differences

plus chance variations (“error”):

Data = fit (“factor/groups”) + residual (“error”)

The one-way ANOVA model analyses

situations where chance variations are

normally distributed N(0,σ) so that:

Testing hypotheses in one-way ANOVAWe have I independent SRSs, from I populations or treatments.

The ith population has a normal distribution with unknown mean µi.

All I populations have the same standard deviation σ, unknown.

The ANOVA F statistic tests:

When H0 is true, F has the F

distribution with I − 1 (numerator)

and N − I (denominator) degrees of

freedom.

H0: = = … = I

Ha: not all the i are equal.)(SSE

)1(SSG

IN

IF

sample samein sindividual amongvariation

means sample amongvariation F

Difference in

means small

relative to overall

variability

Difference in

means large

relative to overall

variability

Larger F-values typically yield more significant results. How large depends on

the degrees of freedom (I − 1 and N − I).

The ANOVA F-statistic compares variation due to specific sources

(levels of the factor) with variation among individuals who should be

similar (individuals in the same sample).

F tends to be small F tends to be large

The ANOVA F-test

Checking our assumptions

The ANOVA F-test requires that all populations have the same

standard deviation . Since is unknown, this can be hard to check.

Practically: The results of the ANOVA F-test are approximately

correct when the largest sample standard deviation is no more than

twice as large as the smallest sample standard deviation.

(Equal sample sizes also make ANOVA more robust to deviations from the equal rule)

Each of the #I populations must be normally distributed (histograms

or normal quantile plots). But the test is robust to normality deviations

for large enough sample sizes, thanks to the central limit theorem.

Do nematodes affect plant growth?

Seedling growth si

0 nematode 10.8 9.1 13.5 9.2 10.65 2.0531000 nematodes 11.1 11.1 8.2 11.3 10.425 1.4865000 nematodes 5.4 4.6 7.4 5.0 5.6 1.24410000 nematodes 5.8 5.3 3.2 7.5 5.45 1.771

Conditions required:

• equal variances: checking that largest si no more than twice smallest si

Largest si = 2.053; smallest si = 1.244

• Independent SRSs

Four groups obviously independent

• Distributions “roughly” normal

It is hard to assess normality with only

four points per condition. But the pots in

each group are identical, and there is no

reason to suspect skewed distributions.

x i

A study of the effect of smoking classifies subjects as nonsmokers, moderate

smokers, and heavy smokers. The investigators interview a random sample of

200 people in each group and ask “How many hours do you sleep on a typical

night?”

1. Study design?

2. Hypotheses?

3. ANOVA assumptions?

4. Degrees of freedom?

1. This is an observational study.Explanatory variable: smoking -- 3 levels: nonsmokers, moderate smokers, heavy smokersResponse variable: # hours of sleep per night

2. H0: all 3 i equal (versus not all equal)

3. Three obviously independent SRS. Sample size of 200 should accommodate any departure from normality. Would still be good to check for smin/smax.

4. I = 3, n1 = n2 = n3 = 200, and N = 600, so there are I - 1 = 2 (numerator) and N - I = 597 (denominator) degrees of freedom.

Smoking influence on sleep

The ANOVA tableSource of variation Sum of squares

SSDF Mean square

MSF P value F crit

Among or between “groups”

I -1 SSG/DFG MSG/MSE Tail area above F

Value of F for

Within groups or “error”

N - I SSE/DFE

Total SST=SSG+SSE N – 1

2)( xxn ii

2)( xxij

2)1( ii sn

R2 = SSG/SST √MSE = sp

Coefficient of determination Pooled standard deviation

The sum of squares represents variation in the data: SST = SSG + SSE.

The degrees of freedom likewise reflect the ANOVA model: DFT = DFG + DFE.

Data (“Total”) = fit (“Groups”) + residual (“Error”)

Excel output for the one-way ANOVA

numeratordenominator

Here, the calculated F-value (12.08) is larger than Fcritical (3.49) for 0.05.

Thus, the test is significant at 5% Not all mean seedling lengths are

the same; the number of nematode’s is an influential factor.

ANOVA

SeedlingLength

100.647 3 33.549 12.080 .001

33.328 12 2.777

133.974 15

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

SPSS output for the one-way ANOVA

The ANOVA found that the amount of

nematodes in pots significantly impacts

seedling growth.

The graph suggests that nematode

amounts above 1,000 per pot are

detrimental to seedling growth.

Using Table EThe F distribution is asymmetrical and has two distinct degrees of

freedom. This was discovered by Fisher, hence the label “F.”

Once again, what we do is calculate the value of F for our sample data

and then look up the corresponding area under the curve in Table E.

dfnum = I − 1

dfden

=N − I

Fp

For df: 5,4

Table E

Fcritical for 5% is 3.49

F = 12.08 > 10.80

Thus p < 0.001

ANOVASource of Variation SS df MS F P-value F critBetween Groups 101 3 33.5 12.08 0.00062 3.4903Within Groups 33.3 12 2.78

Total 134 15

Yogurt preparation and taste

Yogurt can be made using three distinct commercial preparation

methods: traditional, ultra filtration, and reverse osmosis.

To study the effect of these methods on taste, an experiment was

designed where three batches of yogurt were prepared for each of the

three methods. A trained expert tasted each of the nine samples,

presented in random order, and judged them on a scale of 1 to 10.

Variables, hypotheses, assumptions, calculations?

ANOVA table

Source of variation SS df MS F P-value F crit

Between groups 17.3 I-1=2 8.65 11.283

Within groups 4.6 N-I=6 0.767

Total 17.769

dfnum = I − 1

dfden

=N − I F

MSG, the mean square for groups, measures how different the individual

means are from the overall mean (~ weighted average of square distances of

sample averages to the overall mean). SSG is the sum of squares for groups.

MSE, the mean square for error is the pooled sample variance sp2 and

estimates the common variance σ2 of the I populations (~ weighted average of

the variances from each of the I samples). SSE is the sum of squares for error.

)(SSE

)1(SSG

MSE

MSG

IN

IF

Computation details

MSG n1(x 1 x )2 n2(x 2 x )2 L nI (x I x )2

I 1

MSE (n1 1)s1

2 (n2 1)s22 L (nI 1)sI

2

N I

One-way ANOVA Comparing the means

IPS Chapter 12.2

© 2009 W.H. Freeman and Company

Objectives (IPS Chapter 12.2)

Comparing the means

Contrasts

Multiple comparisons

Power of the one-way ANOVA test

You have calculated a p-value for your ANOVA test. Now what?

If you found a significant result, you still need to determine which

treatments were different from which.

You can gain insight by looking back at your plots (boxplot, mean ± s).

There are several tests of statistical significance designed specifically for

multiple tests. You can choose apriori contrasts, or aposteriori multiple

comparisons.

You can find the confidence interval for each mean i shown to be

significantly different from the others.

Contrasts can be used only when there are clear expectations BEFORE starting an experiment, and these are reflected in the experimental design. Contrasts are planned comparisons.

Patients are given either drug A, drug B, or a placebo. The three treatments are not symmetrical. The placebo is meant to provide a baseline against which the other drugs can be compared.

Multiple comparisons should be used when there are no justified expectations. Those are aposteriori, pair-wise tests of significance.

We compare gas mileage for eight brands of SUVs. We have no prior knowledge to expect any brand to perform differently from the rest. Pair-wise comparisons should be performed here, but only if an ANOVA test on all eight brands reached statistical significance first.

It is NOT appropriate to use a contrast test when suggested comparisons appear only after the data is collected.

Contrasts: planned comparisons When an experiment is designed to test a specific hypothesis that

some treatments are different from other treatments, we can use

contrasts to test for significant differences between these specific

treatments.

Contrasts are more powerful than multiple comparisons because they are

more specific. They are more able to pick up a significant difference.

You can use a t-test on the contrasts or calculate a t-confidence interval.

The results are valid regardless of the results of your multiple sample

ANOVA test (you are still testing a valid hypothesis).

A contrast is a combination of

population means of the form :

Where the coefficients ai have sum 0.

The corresponding sample contrast is :

The standard error of c is :

iia

ii xac

cSEct

i

i

i

ipc n

aMSE

n

asSE

22

To test the null hypothesis H0: = 0 use the t-statistic:

With degrees of freedom DFE that is associated with sp. The alternative hypothesis can be one- or two-sided.

cSEtc *

A level C confidence interval for the difference is :

Where t* is the critical value defining the middle C% of the t distribution with DFE degrees of freedom.

Contrasts are not always readily available in statistical software

packages (when they are, you need to assign the coefficients “ai”), or

may be limited to comparing each sample to a control.

If your software doesn’t provide an option for contrasts, you can test

your contrast hypothesis with a regular t-test using the formulas we just

highlighted. Remember to use the pooled variance and degrees of

freedom as they reflect your better estimate of the population variance.

Then you can look up your p-value in a table of t-distribution.

Do nematodes affect plant growth? A botanist prepares

16 identical planting pots and adds different numbers of

nematodes into the pots. Seedling growth

(in mm) is recorded two weeks later.

Nematodes and plant growth

Nematodes0 10.8 9.1 13.5 9.2 10.65

1,000 11.1 11.1 8.2 11.3 10.435,000 5.4 4.6 7.4 5 5.6

10,000 5.8 5.3 3.2 7.5 5.45

Seedling growth

overall mean 8.03

x i

One group contains no nematode at all. If the botanist planned this group as a

baseline/control, then a contrast of all the nematode groups against the

control would be valid.

Nematodes: planned comparison

475.1045.56.5425.1065.10*3 ii xac

Contrast of all the nematode groups against the control:

Combined contrast hypotheses:

H0: µ1 = 1/3 (µ2+ µ3 + µ4) vs.

Ha: µ1 > 1/3 (µ2+ µ3 + µ4) one tailed

Contrast coefficients: (+1 −1/3 −1/3 −1/3) or (+3 −1 −1 −1)

9.24

)1(*3

4

3*78.2

222

i

ipc n

asSE

12 :df 6.39.25.10 N-ISEct c

In Excel: TDIST(3.6,12,1) = tdist(t, df, tails) ≈ 0.002 (p-value).

Nematodes result in significantly shorter seedlings (alpha 1%).

x¯i si

G1: 0 nematode 10.65 2.053G2: 1,000 nematodes 10.425 1.486G3: 5,000 nematodes 5.6 1.244G4: 1,0000 nematodes 5.45 1.771

ANOVA

SeedlingLength

100.647 3 33.549 12.080 .001

33.328 12 2.777

133.974 15

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

Contrast Coefficients

-3 1 1 1Contrast1

0 1000 5000 10000

NematodesLevel

Contrast Tests

-10.4750 2.88650 -3.629 12 .003

-10.4750 3.34823 -3.129 4.139 .034

Contrast1

1

Assume equal variances

Does not assume equalvariances

SeedlingLength

Value ofContrast Std. Error t df Sig. (2-tailed)

Planned comparison:

H0: µ1 = 1/3 (µ2+ µ3 + µ4) vs.

Ha: µ1 > 1/3 (µ2+ µ3 + µ4) one tailed

Contrast coefficients: (+3 −1 −1 −1)

ANOVA: H0: all µi are equal vs. Ha: not all µi are equal

ANOVA vs. contrasts in SPSS

not all µi are equal

Nematodes result in significantly shorter seedlings (alpha 1%).

Multiple comparisonsMultiple comparison tests are variants on the two-sample t-test.

They use the pooled standard deviation sp = √MSE.

The pooled degrees of freedom DFE.

And they compensate for the multiple comparisons.

We compute the t-statistic for all pairs of means:

A given test is significant (µi and µj significantly different), when

|tij| ≥ t** (df = DFE).

The value of t** depends on which procedure you choose to use.

tij (x i x j )

sp

1ni

1n j

The Bonferroni procedure performs a number of pair-wise

comparisons with t-tests and then multiplies each p-value by the

number of comparisons made. This ensures that the probability of

making any false rejection among all comparisons made is no greater

than the chosen significance level α.

As a consequence, the higher the number of pair-wise comparisons you

make, the more difficult it will be to show statistical significance for each

test. But the chance of committing a type I error also increases with the

number of tests made. The Bonferroni procedure lowers the working

significance level of each test to compensate for the increased chance of

type I errors among all tests performed.

The Bonferroni procedure

Simultaneous confidence intervals

We can also calculate simultaneous level C confidence intervals for

all pair-wise differences (µi − µj) between population means:

jipji nn

stxxCI11

**)(:

sp is the pooled variance, MSE.

t** is the t critical with degrees of freedom DFE = N – I, adjusted for

multiple, simultaneous comparisons (e.g., Bonferroni procedure).

SYSTAT File contains variables: GROWTH NEMATODES$Categorical values encountered during processing are:NEMATODES$ (four levels): 10K, 1K, 5K, none

Dep Var: GROWTH N: 16 Multiple R: 0.867 Squared multiple R: 0.751 Analysis of VarianceSource Sum-of-Squares df Mean-Square F-ratio PNEMATODES$ 100.647 3 33.549 12.080 0.001Error 33.328 12 2.777

Post Hoc test of GROWTH Using model MSE of 2.777 with 12 df. Matrix of pairwise mean differences:

1 2 3 41 0.0002 4.975 0.0003 0.150 −4.825 0.0004 5.200 0.225 5.050 0.000

Bonferroni AdjustmentMatrix of pairwise comparison probabilities:

1 2 3 41 1.0002 0.007 1.0003 1.000 0.009 1.0004 0.005 1.000 0.006 1.000

SigmaStat—One-Way Analysis of Variance

Normality Test: Passed (P > 0.050)Equal Variance Test: Passed (P = 0.807)

Group Name N Missing Mean Std dev SEMNone 4 0 10.650 2.053 1.0271K 4 0 10.425 1.486 0.7435K 4 0 5.600 1.244 0.62210K 4 0 5.450 1.771 0.886

Source of variation DF SS MS F P Between groups 3 100.647 33.549 12.080 <0.001Residual 12 33.328 2.777Total 15 133.974

Power of performed test with alpha = 0.050: 0.992

All Pairwise Multiple Comparison Procedures (Bonferroni t-test):Comparisons for factor: NematodesComparison Diff of means t P P<0.050None vs. 10K 5.200 4.413 0.005 YesNone vs. 5K 5.050 4.285 0.006 YesNone vs. 1K 0.225 0.191 1.000 No1K vs. 10K 4.975 4.222 0.007 Yes1K vs. 5K 4.825 4.095 0.009 Yes5K vs. 10K 0.150 0.127 1.000 No

Multiple Comparisons

Dependent Variable: SeedlingLength

Bonferroni

.22500 1.17841 1.000 -3.4901 3.9401

5.05000* 1.17841 .006 1.3349 8.7651

5.20000* 1.17841 .005 1.4849 8.9151

-.22500 1.17841 1.000 -3.9401 3.4901

4.82500* 1.17841 .009 1.1099 8.5401

4.97500* 1.17841 .007 1.2599 8.6901

-5.05000* 1.17841 .006 -8.7651 -1.3349

-4.82500* 1.17841 .009 -8.5401 -1.1099

.15000 1.17841 1.000 -3.5651 3.8651

-5.20000* 1.17841 .005 -8.9151 -1.4849

-4.97500* 1.17841 .007 -8.6901 -1.2599

-.15000 1.17841 1.000 -3.8651 3.5651

(J) NematodesLevel1000

5000

10000

0

5000

10000

0

1000

10000

0

1000

5000

(I) NematodesLevel0

1000

5000

10000

MeanDifference

(I-J) Std. Error Sig. Lower Bound Upper Bound

95% Confidence Interval

The mean difference is significant at the .05 level.*.

SPSSANOVA

SeedlingLength

100.647 3 33.549 12.080 .001

33.328 12 2.777

133.974 15

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

The ANOVA test is significant (α 5%): we have found evidence that the three

methods do not all yield the same population mean reading comprehension score.

Teaching methods

A study compares the reading

comprehension (“COMP,” a test

score) of children randomly

assigned to one of three teaching

methods: basal, DRTA, and

strategies.

We test: H0: µBasal = µDRTA = µStrat vs. Ha: H0 not true

What do you conclude?

The three methods do not yield the same results: We found evidence of a

significant difference between DRTA and basal methods (DRTA gave better

results on average), but the data gathered does not support the claim of a

difference between the other methods (DRTA vs. strategies or basal vs.

strategies).

PowerThe power, or sensitivity, of a one-way ANOVA is the probability that

the test will be able to detect a difference among the groups (i.e. reach

statistical significance) when there really is a difference.

Estimate the power of your test while designing your experiment to

select sample sizes appropriate to detect an amount of difference

between means that you deem important.

Too small a sample is a waste of experiment, but too large a sample is

also a waste of resources.

A power of at least 80% is often suggested.

Power computations

ANOVA power is affected by

The significance level

The sample sizes and number of groups being compared

The differences between group means µi

The guessed population standard deviation

You need to decide what alternative Ha you would consider important,

detect statistically for the means µi, and to guess the common standard

deviation σ (from similar studies or preliminary work).

The power computations then require calculating a non-centrality

paramenter λ, which follows the F distribution with DFG and DFE

degrees of freedom to arrive at the power of the test.

Systat: Power analysis Alpha = 0.05

Model = Oneway

Number of groups = 4

Within cell S.D. = 1.5

Mean(01) = 7.0

Mean(02) = 8.0

Mean(03) = 9.0

Mean(04) = 10.0

Effect Size = 0.745

Noncentrality parameter = 2.222 * sample size

SAMPLE SIZE POWER

(per cell)

3 0.373

4 0.551

5 0.695

6 0.802

7 0.876

8 0.925

If we anticipated a gradual decrease of

seedling length for increasing amounts

of nematodes in the pots and would

consider gradual changes of 1 mm on

average to be important enough to be

reported in a scientific journal…

…then we would reach a power of 80%

or more when using six pots or more for

each condition.

(Four pots per condition would only

bring a power of 55%.)

guessed

Systat: Power analysis Alpha = 0.05

Power = 0.80

Model = One-way

Number of groups = 4

Within cell S.D. = 1.7

Mean(01) = 6.0

Mean(02) = 6.0

Mean(03) = 10.0

Mean(04) = 10.0

Effect size = 1.176

Noncentrality parameter = 5.536 * sample size

SAMPLE SIZE POWER (per cell)

2 0.394

3 0.766

4 0.931

Total sample size = 16

If we anticipated that only large amounts

of nematodes in the pots would lead to

substantially shorter seedling lengths

and would consider step changes of 4

mm on average an important effect…

…then we would reach a power of 80%

or more when using four pots or more in

each group (three pots per condition

might be close enough though).

guessed